Pesin Theory Research Articles

Pathology and asymmetry: Centralizer rigidity for partially hyperbolic diffeomorphisms

We discover a rigidity phenomenon within the volume-preserving partially hyperbolic diffeomorphisms with 1-dimensional center. In particular, for smooth ergodic perturbations of certain algebraic systems—including the discretized geodesic flows over hyperbolic manifolds and certain toral automorphisms with simple spectrum and exactly one eigenvalue on the unit circle—the smooth centralizer is either virtually Zl or contains a smooth flow. At the heart of this work are two very different rigidity phenomena. The first was discovered by Avila, Viana, and the second author: for a class of volume-preserving partially hyperbolic systems including those studied here, the disintegration of volume along the center foliation is equivalent either to Lebesgue or atomic. The second phenomenon, described by the first and third authors, is the rigidity associated to several commuting partially hyperbolic diffeomorphisms with very different hyperbolic behavior transverse to a common center foliation. We employ a variety of techniques, among them a novel geometric approach to building new partially hyperbolic elements in hyperbolic Weyl chambers using Pesin theory and leafwise conjugacy, measure rigidity via thermodynamic formalism for circle extensions of Anosov diffeomorphisms, partially hyperbolic Livsic theory, and nonstationary normal forms.

SINKS AND SOURCES FOR C1 DYNAMICS WHOSE LYAPUNOV EXPONENTS HAVE CONSTANT SIGN

Let $f:M\to M$ be a $C^1$ map of a compact manifold $M$, with dimension at least $2$, admitting some point whose future trajectory has only negative Lyapunov exponents. Then this trajectory converges to a periodic sink. We need only assume that $Df$ is never the null map at any point (in particular, we need no extra smoothness assumption on $Df$ nor the existence of a invariant probability measure), encompassing a wide class of possible critical behavior. Similarly, a trajectory having only positive Lyapunov exponents for a $C^1$ diffeomorphism is itself a periodic repeller (source). Analogously for a $C^1$ open and dense subset of vector field on finite dimensional manifolds: for a flow $\phi_t$ generated by such a vector field, if a trajectory admits weak asymptotic sectional contraction (the extreme rates of expansion of the Linear Poincare Flow are all negative), then this trajectory belongs either to the basin of attraction of a periodic hyperbolic attracting orbit (a periodic sink or an attracting equilibrium); or the trajectory accumulates a codimension one saddle singularity. Similar results hold for weak sectional expanding trajectories. Both results extend part of the non-uniform hyperbolic theory (Pesin's Theory) from the $C^{1+}$ diffeomorphism setting to $C^1$ endomorphisms and $C^1$ flows. Some ergodic theoretical consequences are discussed. The proofs use versions of Pliss' Lemma for maps and flows translated as (reverse) hyperbolic times, and a result ensuring that certain subadditive cocycles over $C^1$ vector fields are in fact additive.

On mixing diffeomorphisms of the disc

We prove that a $$C^k$$, $$k\ge 2$$ pseudo-rotation f of the disc with non-Brjuno rotation number is $$C^{k-1}$$-rigid. The proof is based on two ingredients: (1) we derive from Franks’ Lemma on free discs that a pseudo-rotation with small rotation number compared to its $$C^1$$ norm must be close to the identity map; (2) using Pesin theory, we obtain an effective finite information version of the Katok closing lemma for an area preserving surface diffeomorphism f, that provides a controlled gap in the possible growth of the derivatives of f between exponential and sub-exponential. Our result on rigidity, together with a KAM theorem by Russmann, allow to conclude that analytic pseudo-rotations of the disc or the sphere are never topologically mixing. Due to a structure theorem by Franks and Handel of zero entropy surface diffeomorphisms, it follows that an analytic conservative diffeomorphism of the disc or the sphere that is topologically mixing must have positive topological entropy.

Dominated Pesin theory: convex sum of hyperbolic measures

In the uniformly hyperbolic setting it is well known that the set of all measures supported on periodic orbits is dense in the convex space of all invariant measures. In this paper we consider the converse question, in the non-uniformly hyperbolic setting: assuming that some ergodic measure converges to a convex combination of hyperbolic ergodic measures, what can we deduce about the initial measures? To every hyperbolic measure μ whose stable/unstable Oseledets splitting is dominated we associate canonically a unique class H(μ) of periodic orbits for the homoclinic relation, called its intersection class. In a dominated setting, we prove that a measure for which almost every measure in its ergodic decomposition is hyperbolic with the same index, such as the dominated splitting, is accumulated by ergodic measures if, and only if, almost all such ergodic measures have a common intersection class. We provide examples which indicate the importance of the domination assumption.

Quantitative Pesin theory for Anosov diffeomorphisms and flows

Pesin sets are measurable sets along which the behavior of a matrix cocycle above a measure-preserving dynamical system is explicitly controlled. In uniformly hyperbolic dynamics, we study how often points return to Pesin sets under suitable conditions on the cocycle: if it is locally constant, or if it admits invariant holonomies and is pinching and twisting, we show that the measure of points that do not return a linear number of times to Pesin sets is exponentially small. We discuss applications to the exponential mixing of contact Anosov flows and consider counterexamples illustrating the necessity of suitable conditions on the cocycle.

Large-deviation properties in some non-uniformly hyperbolic systems via Pesin theory

ABSTRACTThis article is devoted to level-1 large-deviation properties in some non-uniformly hyperbolic systems via Pesin theory. In particular, our result can be applied to the diffeomorphisms on surfaces.

Real bounds and Lyapunov exponents

We prove that a $C^3$ critical circle map without periodic points has zero Lyapunov exponentwith respect to its unique invariant Borel probability measure. Moreover, no critical point of such a mapsatisfies the Collet-Eckmann condition. This result is proved directly from the well-known real a-priori bounds,without using Pesin's theory. We also show how our methods yield an analogous result for infinitely renormalizableunimodal maps of any combinatorial type. Finally we discuss an application of these facts to the study of neutralmeasures of certain rational maps of the Riemann sphere.

Pesin theory and equilibrium measures on the interval

We use Pesin theory to study possible equilibrium measures for piecewise monotone maps of the interval. The maps may have unbounded derivative.

On stability and hyperbolicity for polynomial automorphisms of C^2

Let (fλ)λ∈Λ be a holomorphic family of polynomial automorphisms of C2. Fol- lowing previous work of Dujardin and Lyubich, we say that such a family is weakly stable if saddle periodic orbits do not bifurcate. It is an open question whether this property is equivalent to structural stability on the Julia set J∗ (that is, the closure of the set of saddle periodic points). In this paper we introduce a notion of regular point for a polynomial automorphism, inspired by Pesin theory, and prove that in a weakly stable family, the set of regular points moves holomorphically. It follows that a weakly stable family is probabilistically structurally stable, in a very strong sense. Another consequence of these techniques is that weak stability preserves uniform hyperbolicity on J∗.

On cusps and flat tops

We develop non-invertible Pesin theory for a new class of maps called cusp maps. These maps may have unbounded derivative, but nevertheless verify a property analogous to $C^{1+\epsilon}$. We do not require the critical points to verify a non-flatness condition, so the results are applicable to $C^{1+\epsilon}$ maps with flat critical points. If the critical points are too flat, then no absolutely continuous invariant probability measure can exist. This generalises a result of Benedicks and Misiurewicz.

The $C^{1+\alpha }$ hypothesis in Pesin Theory revisited

We show that for every compact 3-manifold $M$ there exists an open subset of $\diff ^1(M)$ in which every generic diffeomorphism admits uncountably many ergodic probability measures which are hyperbolic while their supports are disjoint and admit a basis of attracting neighborhoods and a basis of repelling neighborhoods. As a consequence, the points in the support of these measures have no stable and no unstable manifolds. This contrasts with the higher regularity case, where Pesin theory gives us the stable and the unstable manifolds with complementary dimensions at almost every point. We also give such an example in dimension two, without local genericity.

SRB attractors with intermingled basins for non-hyperbolic diffeomorphisms

We investigate a class of non-hyperbolic diffeomorphisms defined onthe product space. By using the Pesin theory combined with thegeneral theory of differentiable dynamical systems, we prove thatthere are exactly two SRB attractors, and their basins cover a fullmeasure subset of the ambient manifold. Furthermore, we prove that the basins ofSRB attractors have the strange intermingled phenomenon, i.e. theyare measure-theoretically dense in each other. The intermingledphenomena have been observed in many physical systems by numericalexperiments, and considered to be important to some fundamentalproblems in physical, biology and computer science etc. Finally, wedescribe a concrete example for application.

Ergodicity of the Geodesic Flow on Non-complete Negatively Curved Surfaces

We study the dynamics of the geodesic flow for a class of non-complete Riemannian metrics on a negatively curved surface M . These finite area surfaces are composed of finitely many “singular” surfaces of revolutions of the form y = x, r > 1 for 0 ≤ x ≤ 1 (thin pieces), together with connecting surfaces of bounded negative curvature (thick pieces). The curvature at every point is negative and bounded away from zero, but is unbounded in the “cusps.” The geodesic flow is non-complete because geodesics corresponding to singular unit tangent vectors pointing into the origin hit the cusps in finite time and then cease to be defined. Such singular unit tangent vectors are dense in the unit tangent bundle. Our main result is ergodicity of the geodesic flow (Theorem 5.1). It immediately follows that these metrics have dense geodesics in the unit tangent bundle SM . We also prove that the closed geodesics are dense in the unit tangent bundle (Theorem 6.1) The motivation for considering these metrics arises from studying the geodesic flow for the Weil-Petersson (WP) metric on two dimensional moduli spaces of Riemann surfaces., e.g., the moduli space for the once punctured torus. In several fundamental ways the geodesic flow we consider provides a good model for the WP geodesic flow. For example, Wolpert showed that the WP metric is also non-complete, has finite area, has negative curvature bounded away from zero, and is unbounded in the ”cusps.” Also, various authors have shown that the cusps can be approximated by singular surface of revolutions. At present, there is only a single technical obstruction (additional estimates on the derivatives of the GF in the cusp) that prevent us from applying this general method to prove ergodicity of the WP geodesic flow. See Section 7. The geometric properties of both types of flows imply that the geodesic flow is a uniformly hyperbolic dynamical system with singularities. The non-completeness causes pathologies in the stable and unstable manifolds, as for many billiard flows. For example, stable and unstable manifolds at a point, if they exist, may intersect a cusp point, and thus have only finite length. One needs extensions of “Pesin theory” to systems with singularities to study these geodesic flows. Another challenging aspect is that these surfaces may be simply connected, making the use of any of the traditional arguments involving boundaries of the covering spaces inapplicable. Studying the dynamics requires the development of new approches. Our strategy to prove ergodicity of the geodesic flow is to first establish non-uniform hyperbolicity, i.e., the Lyapunov exponents are non-zero at almost every point. We then prove that there

Pointwise dimension and ergodic decompositions

We study the Hausdorff dimension and the pointwise dimension of measures that are not necessarily ergodic. In particular, for conformal expanding maps and hyperbolic diffeomorphisms on surfaces we establish explicit formulas for the pointwise dimension of an arbitrary invariant measure in terms of the local entropy and of the Lyapunov exponents. These formulas are obtained with a direct approach, pertaining to the fundamentals of ergodic theory, which in particular does not require Pesin theory. This allows us to show, for those systems, that the Hausdorff dimension of a (non-ergodic) invariant measure is equal to the essential supremum of the Hausdorff dimensions of the measures in an ergodic decomposition. We also establish corresponding results for non-uniformly hyperbolic diffeomorphisms on surfaces, now using Pesin theory. To the best of our knowledge, the formula for the Hausdorff dimension of an invariant measure along an ergodic decomposition is new even in the case of uniformly hyperbolic dynamics.

Invariant measure and Lyapunov exponents for birational maps of P^2

In this paper we construct and study a natural invariant measure for a birational self-map of the complex projective plane. Our main hypothesis - that the birational map be "separating" - is a condition on the indeterminacy set of the map. We prove that the measure is mixing and that it has distinct Lyapunov exponents. Under a further hypothesis on the indeterminacy set we show that the measure is hyperbolic in the sense of Pesin theory. In this case, we also prove that saddle periodic points are dense in the support of the measure.

The C1+α hypothesis in Pesin theory

The C1+α hypothesis in Pesin theory

Characteristic Exponents and Invariant Manifolds in Hilbert Space

The multiplicative ergodic theorem and the construction almost everywhere of stable and unstable manifolds (Pesin theory) are extended to differentiable dynamical systems on Hilbert manifolds under some compactness assumptions. The results apply to partial differential equations of evolution and also to non-invertible maps of compact manifolds.